Pre-Calc 12

5.2 Transformations of Sinusoidal

Functions

Big Idea:

Understanding the characteristics of families of functions allows us to model and understand

relationships and to build connections between classes of functions.

Curricular Competencies:

Explain and justify math ideas and decisions

Visualize to explore math

Vertical Displacement and Phase Shift

For periodic functions, a vertical translation is called a vertical displacement, while a horizontal

translation is called a phase shift.

Example 1: Sketch the graph of 𝑦 = sin(𝑥 − 30°) + 1 for at least one cycle.

Vertical displacement: Period: Amplitude:

Phase shift: Domain: Range:

Pre-Calc 12

Example 2: Sketch the graph of 𝑦 = −cos(𝑥 + 𝜋) − 1 for at least one cycle.

Vertical displacement: Period: Amplitude:

Phase shift: Domain: Range:

Transformation connection ….

𝒚 = 𝒂𝒇(𝒃(𝒙 − 𝒉)) + 𝒌 𝒚 = 𝒂𝒔𝒊𝒏(𝒃(𝒙 − 𝒄)) + 𝒅

Pre-Calc 12

Example 3: Sketch the graph of 𝑦 = 3sin (2𝑥 −2𝜋

3) + 2 for at least one cycle.

Vertical displacement: Period: Amplitude:

Phase shift: Domain: Range:

Example 4: Sketch the graph of 𝑦 = −2cos𝜋

6(𝑥 + 3) − 1 for at least one cycle.

Vertical displacement: Period: Amplitude:

Phase shift: Domain: Range:

Pre-Calc 12

Equation of 𝑦 = 𝑎𝑠𝑖𝑛𝑏(𝑥 − 𝑐) + 𝑑 or 𝑦 = 𝑎𝑐𝑜𝑠𝑏(𝑥 − 𝑐) + 𝑑

𝒂 =𝒚𝒎𝒂𝒙−𝒚𝒎𝒊𝒏

𝟐 𝒃 =

𝟐𝝅

𝑷 𝒄 = starting point 𝒅 = midline

Example 5: Write sinusoidal equations of the form 𝑦 = 𝑎𝑠𝑖𝑛𝑏(𝑥 − 𝑐) + 𝑑 f and 𝑦 = 𝑎𝑐𝑜𝑠𝑏(𝑥 − 𝑐) + 𝑑

to represent the function shown in the graph.

Example 6: Prince Rupert, British Columbia, has the deepest natural harbor in North America. The

depth, d, in meters, of the berths for the ships can be approximated by the equation 𝑑(𝑡) = 8𝑐𝑜𝑠𝜋

6𝑡 +

12, where 𝑡 is the time, in hours, after the first high tide.

a) Using your graphing calculator, graph the function over 2 cycles.

b) What is the period of the tide?

c) An ocean liner requires a minimum of 13 m

of water to dock safely. Determine the

number of hours per cycle the ocean liner

can safely dock.

d) If the minimum depth of the berth occurs

at 6 h, determine the depth of the water.

At what other times is the water level at a

Minimum?

Assignment: p 250 1 acef, 2 acef, 3-5, 6ac, 7ac, 9, 10, 13, 14-16, 20, 26

Hint … 5280 ft = 1 mi